On Lawrence–Krammer representations

Lawrence–Krammer representations (or simply LK-representations for short) are linear representations of Artin–Tits groups of small type over the ring Z[x±1,y±1], which are of particular interest since they are known to be faithful when the type is spherical, and irreducible when the type is spherical and connected.Here, we define and study LK-representations over an arbitrary commutative ring R. Under some conditions on R and on the parameters of an LK-representation, which generalize the classical settings, we recover the faithfulness property and give a new proof of irreducibility that works for every (non-necessarily spherical) connected type. We then study the equivalence of distinct LK-representations of a given Artin–Tits group, for distinct choices of parameters. Finally, we generalize to our setting the known classification of LK-representations in the spherical cases, and proceed with their classification in the affine cases.